Mordell-Weil lattices and toric decompositions of plane curves

We extend results of Cogolludo-Agustin and Libgober relating the Alexander polynomial of a plane curve C with the Mordell-Weil rank of certain isotrivial families of jacobians over of discriminant C. In the second part we introduce a height pairing on the (2, 3, 6) quasi-toric decompositions of a plane curve. We use this pairing and the results in the first part of the paper to construct a pair of degree 12 curves with 30 cusps and Alexander polynomial , but with distinct height pairing. We use the height pairing to show that these curves from a Zariski pair.